The Disjoint Domination Game

We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game is started by the breaker. This implies the same in the $(2:1)$ biased game also in the maker-start game. It remains open to characterize the maker-win graphs in the maker-start non-biased game, and to analyze the $(a:b)$ biased game for $(a:b)\neq (2:1)$. For a more restricted variant of the non-biased game we prove that the maker can win on every graph without isolated vertices.Comment: 18 page

Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

Pareto optimality in house allocation problems

We study Pareto optimal matchings in the context of house allocation problems. We present an O(\sqrt{n}m) algorithm, based on Gales Top Trading Cycles Method, for finding a maximum cardinality Pareto optimal matching, where n is the number of agents and m is the total length of the preference lists. By contrast, we show that the problem of finding a minimum cardinality Pareto optimal matching is NP-hard, though approximable within a factor of 2. We then show that there exist Pareto optimal matchings of all sizes between a minimum and maximum cardinality Pareto optimal matching. Finally, we introduce the concept of a signature, which allows us to give a characterization, checkable in linear time, of instances that admit a unique Pareto optimal matching

Maker-Breaker domination game on trees when Staller wins

In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $\gamma_{\rm SMB}(G)$ (resp., $\gamma_{\rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $\gamma_{\rm SMB}'(T)=k$ are characterized. Applying hypergraphs, a general upper bound on $\gamma_{\rm SMB}'$ is proved. Let $S = S(n_1,\dots, n_\ell)$ be the subdivided star obtained from the star with $n$ edges by subdividing its edges $n_1-1, \ldots, n_\ell-1$ times, respectively. Then $\gamma_{\rm SMB}'(S)$ is determined in all the cases except when $\ell\ge 4$ and each $n_i$ is even. The simplest formula is obtained when there are are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $\gamma_{\rm SMB}'(S(n_1,\dots, n_\ell))=\lceil \log_2(n_1+n_2+1)\rceil$. For caterpillars, exact formulas for $\gamma_{\rm SMB}$ and for $\gamma_{\rm SMB}'$ are established

On the Parameterized Complexity of [1,j]-Domination Problems

For a graph G, a set D subseteq V(G) is called a [1,j]-dominating set if every vertex in V(G) setminus D has at least one and at most j neighbors in D. A set D subseteq V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study [1,j]-Dominating Set on sparse graph classes from the perspective of parameterized complexity and prove the following results when the problem is parameterized by solution size: - [1,j]-Dominating Set is W[1]-hard on d-degenerate graphs for d = j + 1; - [1,j]-Dominating Set is FPT on nowhere dense graphs. We also prove that the known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH). Finally, assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth

Optimal pebbling and rubbling of graphs with given diameter

A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $\pi_{opt}$ is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. A rubbling move is similar to a pebbling move, but it can remove the two pebbles from two different vertex. The optimal rubbling number $\rho_{opt}$ is defined analogously to the optimal pebbling number. In this paper we give lower bounds on both the optimal pebbling and rubbling numbers by the distance $k$ domination number. With this bound we prove that for each $k$ there is a graph $G$ with diameter $k$ such that $\rho_{opt}(G)=\pi_{opt}(G)=2^k$

The kk-visibility Localization Game

We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number $\zeta_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded $\zeta_k$ values. Extending results known for the localization number, we show that for $k\geq 2$, every tree contains a subdivision with $\zeta_k = 1$. For many $n$, we give the exact value of $\zeta_k$ for the $n \times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that $\zeta_i \neq \zeta_j$ for all distinct choices of $i$ and $j.